# Why do Maxwell's equations bear his name?

Maxwell's equations in their modern differential form are:

$\nabla \cdot \mathbf{E} = \dfrac {\rho} {\varepsilon_0}$ (Gauss's law for electricity)

$\nabla \cdot \mathbf{B} = 0$ (Gauss's law for magnetism)

$\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}} {\partial t}$ (Maxwell–Faraday equation)

$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)$ (Ampère's circuital law)

I'm aware that Maxwell generalized Faraday's law and Ampère's law by adding displacement current, but why are these four equations named after him although they were discovered by three other people: Gauss, Faraday and Ampère?

It seems to me that his addition to the laws (two of them) are not significant enough to make it bear his name.

• Another question is: Why is the last equation named after Ampère, when he never wrote it down nor did he ever deal with the field concept (cf. this book). Perhaps it's just because it involves current? – Geremia Oct 18 '15 at 5:01

As you noticed, separate equations have other names as well. Maxwell's adding the displacement term made the system complete, with all important consequences, in particular, existing of electromagnetic waves. So the name of the whole system after Maxwell is completely justified.

Ampère never wrote down what is confusingly called "Ampère's circuital law," not even the form without the displacement current term, as Ampère never dealt with the field concept.* Maxwell derived

$$\nabla \times \mathbf{B} = \mu_0\mathbf{J}$$

in his 1855 paper On Faraday's Lines of Force, based on analogies to hydrodynamics, which he corrected to be

$$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)\qquad(2)$$

in his 1861 paper On Physical Lines of Force; he never wrote down Ampère's force law in either paper.

Ampère's force law is completely different from any of Maxwell's equations. It gives the force that current elements $I_1 d\vec {\ell }_1$ and $I_2 d\vec {\ell }_2$ exert on one another to be:

$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$

Thus, it is appropriate that Equation (1) is one of Maxwell's equations. Gauss and Faraday utilized the field concept, so Equation (1) is the most "Maxwellian" of the four Maxwell's equations.

*cf. Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015). Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience (PDF). Montreal: Apeiron. ISBN 978-1-987980-03-5. ch. 15 pp. 221ff.

Let's consider each law seperatey.

1) and 2): Gauss law for electricity and magnetism:

Its integral form was first formulated by Joseph-Louis Lagrange in 1773 (See here).

The next step, the Gauss divergence theorem, was also at first formulated by Joseph-Louis Lagrange in 1762 (See here)

However, it may seem strange, that the credit to both the above equations is given to Gauss.